Use Strategy C25 from Unit C4 to find the equation of the quadric x² + 3y² +52²-8xy-8yz + 3y +3z-6=0 in standard form. Which of the six types of non-degenerate quadric is this quadric?
Use Strategy C25 from Unit C4 to find the equation of the quadric x² + 3y² +52²-8xy-8yz + 3y +3z-6=0 in standard form. Which of the six types of non-degenerate quadric is this quadric?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Use Strategy C25 from Unit C4 to find the equation of the quadric
x² + 3y² +52²-8xy-8yz + 3y +3z-6=0
in standard form. Which of the six types of non-degenerate quadric is
this quadric?

Transcribed Image Text:Strategy C25
To write the quadric with equation
Ax² + By² + Cz²+Fxy + Gyz + Hxz+Jx+Ky+Lz + M = 0
in standard form, do the following.
1. Introduce matrices:
write down the matrices
AF H
}F BG
HG C
A =
and J =
2. Align the axes:
• orthogonally diagonalise A to get
A₁0 0
0 20
0 0 A3/
PT AP =
J
K
L
find (f g h) = JTP, and write the quadric in the form
A₁ (x¹)² + A₂(y)² + A3(2')² + fx' + gy' + hz' + M = 0.
3. Translate the origin:
complete the squares
• make a substitution to change to the coordinate system
(x", y", z").
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